Integrand size = 25, antiderivative size = 977 \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=-24 a b^2 m n^2 x+36 b^3 m n^3 x-12 b^2 m n^2 (a-b n) x+\frac {12 b^2 \sqrt {e} m n^2 (a-b n) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-36 b^3 m n^2 x \log \left (c x^n\right )+\frac {12 b^3 \sqrt {e} m n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )}{\sqrt {f}}+12 b m n x \left (a+b \log \left (c x^n\right )\right )^2-2 m x \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {3 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {6 b^2 \sqrt {-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {3 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {6 b^2 \sqrt {-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {3 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {6 i b^3 \sqrt {e} m n^3 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {6 i b^3 \sqrt {e} m n^3 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {6 b^3 \sqrt {-e} m n^3 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {6 b^2 \sqrt {-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {6 b^3 \sqrt {-e} m n^3 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {6 b^2 \sqrt {-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {6 b^3 \sqrt {-e} m n^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {6 b^3 \sqrt {-e} m n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}} \]
6*b^3*m*n^3*polylog(3,-x*f^(1/2)/(-e)^(1/2))*(-e)^(1/2)/f^(1/2)-6*b^3*m*n^ 3*polylog(3,x*f^(1/2)/(-e)^(1/2))*(-e)^(1/2)/f^(1/2)+6*b^3*m*n^3*polylog(4 ,-x*f^(1/2)/(-e)^(1/2))*(-e)^(1/2)/f^(1/2)-6*b^3*m*n^3*polylog(4,x*f^(1/2) /(-e)^(1/2))*(-e)^(1/2)/f^(1/2)-2*m*x*(a+b*ln(c*x^n))^3+6*a*b^2*n^2*x*ln(d *(f*x^2+e)^m)+6*b^3*n^2*x*ln(c*x^n)*ln(d*(f*x^2+e)^m)-3*b*n*x*(a+b*ln(c*x^ n))^2*ln(d*(f*x^2+e)^m)+36*b^3*m*n^3*x-6*b^3*n^3*x*ln(d*(f*x^2+e)^m)+x*(a+ b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m)+6*I*b^3*m*n^3*polylog(2,I*x*f^(1/2)/e^(1/ 2))*e^(1/2)/f^(1/2)+3*b*m*n*(a+b*ln(c*x^n))^2*ln(1-x*f^(1/2)/(-e)^(1/2))*( -e)^(1/2)/f^(1/2)-3*b*m*n*(a+b*ln(c*x^n))^2*ln(1+x*f^(1/2)/(-e)^(1/2))*(-e )^(1/2)/f^(1/2)+m*(a+b*ln(c*x^n))^3*ln(1+x*f^(1/2)/(-e)^(1/2))*(-e)^(1/2)/ f^(1/2)-m*(a+b*ln(c*x^n))^3*ln(1-x*f^(1/2)/(-e)^(1/2))*(-e)^(1/2)/f^(1/2)- 6*b^2*m*n^2*(a+b*ln(c*x^n))*polylog(2,-x*f^(1/2)/(-e)^(1/2))*(-e)^(1/2)/f^ (1/2)+3*b*m*n*(a+b*ln(c*x^n))^2*polylog(2,-x*f^(1/2)/(-e)^(1/2))*(-e)^(1/2 )/f^(1/2)+6*b^2*m*n^2*(a+b*ln(c*x^n))*polylog(2,x*f^(1/2)/(-e)^(1/2))*(-e) ^(1/2)/f^(1/2)-3*b*m*n*(a+b*ln(c*x^n))^2*polylog(2,x*f^(1/2)/(-e)^(1/2))*( -e)^(1/2)/f^(1/2)-6*b^2*m*n^2*(a+b*ln(c*x^n))*polylog(3,-x*f^(1/2)/(-e)^(1 /2))*(-e)^(1/2)/f^(1/2)+6*b^2*m*n^2*(a+b*ln(c*x^n))*polylog(3,x*f^(1/2)/(- e)^(1/2))*(-e)^(1/2)/f^(1/2)+12*b^2*m*n^2*(-b*n+a)*arctan(x*f^(1/2)/e^(1/2 ))*e^(1/2)/f^(1/2)+12*b^3*m*n^2*arctan(x*f^(1/2)/e^(1/2))*ln(c*x^n)*e^(1/2 )/f^(1/2)-6*I*b^3*m*n^3*polylog(2,-I*x*f^(1/2)/e^(1/2))*e^(1/2)/f^(1/2)...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2302\) vs. \(2(977)=1954\).
Time = 0.46 (sec) , antiderivative size = 2302, normalized size of antiderivative = 2.36 \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\text {Result too large to show} \]
(-2*a^3*Sqrt[f]*m*x + 12*a^2*b*Sqrt[f]*m*n*x - 36*a*b^2*Sqrt[f]*m*n^2*x + 48*b^3*Sqrt[f]*m*n^3*x + 2*a^3*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 6*a ^2*b*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 12*a*b^2*Sqrt[e]*m*n^2*ArcT an[(Sqrt[f]*x)/Sqrt[e]] - 12*b^3*Sqrt[e]*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 6*a^2*b*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] + 12*a*b^2*Sqrt[ e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 12*b^3*Sqrt[e]*m*n^3*ArcTan[ (Sqrt[f]*x)/Sqrt[e]]*Log[x] + 6*a*b^2*Sqrt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqr t[e]]*Log[x]^2 - 6*b^3*Sqrt[e]*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 - 2*b^3*Sqrt[e]*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^3 - 6*a^2*b*Sqrt[ f]*m*x*Log[c*x^n] + 24*a*b^2*Sqrt[f]*m*n*x*Log[c*x^n] - 36*b^3*Sqrt[f]*m*n ^2*x*Log[c*x^n] + 6*a^2*b*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] - 12*a*b^2*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 12*b^3*Sq rt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] - 12*a*b^2*Sqrt[e]*m*n* ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] + 12*b^3*Sqrt[e]*m*n^2*ArcTa n[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] + 6*b^3*Sqrt[e]*m*n^2*ArcTan[(Sqr t[f]*x)/Sqrt[e]]*Log[x]^2*Log[c*x^n] - 6*a*b^2*Sqrt[f]*m*x*Log[c*x^n]^2 + 12*b^3*Sqrt[f]*m*n*x*Log[c*x^n]^2 + 6*a*b^2*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/S qrt[e]]*Log[c*x^n]^2 - 6*b^3*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c *x^n]^2 - 6*b^3*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n]^ 2 - 2*b^3*Sqrt[f]*m*x*Log[c*x^n]^3 + 2*b^3*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)...
Time = 1.55 (sec) , antiderivative size = 988, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2818, 6, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx\) |
| \(\Big \downarrow \) 2818 |
| \(\displaystyle -2 f m \int \left (\frac {6 n^2 x^2 \log \left (c x^n\right ) b^3}{f x^2+e}-\frac {6 n^3 x^2 b^3}{f x^2+e}+\frac {6 a n^2 x^2 b^2}{f x^2+e}-\frac {3 n x^2 \left (a+b \log \left (c x^n\right )\right )^2 b}{f x^2+e}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{f x^2+e}\right )dx+6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle -2 f m \int \left (\frac {6 n^2 x^2 \log \left (c x^n\right ) b^3}{f x^2+e}-\frac {3 n x^2 \left (a+b \log \left (c x^n\right )\right )^2 b}{f x^2+e}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{f x^2+e}+\frac {\left (6 a b^2 n^2-6 b^3 n^3\right ) x^2}{f x^2+e}\right )dx+6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -6 n^3 x \log \left (d \left (f x^2+e\right )^m\right ) b^3+6 n^2 x \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) b^3+6 a n^2 x \log \left (d \left (f x^2+e\right )^m\right ) b^2-3 n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (f x^2+e\right )^m\right ) b+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (f x^2+e\right )^m\right )-2 f m \left (-\frac {18 n^3 x b^3}{f}+\frac {18 n^2 x \log \left (c x^n\right ) b^3}{f}-\frac {6 \sqrt {e} n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right ) b^3}{f^{3/2}}+\frac {3 i \sqrt {e} n^3 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) b^3}{f^{3/2}}-\frac {3 i \sqrt {e} n^3 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right ) b^3}{f^{3/2}}-\frac {3 \sqrt {-e} n^3 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{f^{3/2}}+\frac {3 \sqrt {-e} n^3 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{f^{3/2}}-\frac {3 \sqrt {-e} n^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{f^{3/2}}+\frac {3 \sqrt {-e} n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{f^{3/2}}+\frac {12 a n^2 x b^2}{f}+\frac {6 n^2 (a-b n) x b^2}{f}-\frac {6 \sqrt {e} n^2 (a-b n) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) b^2}{f^{3/2}}+\frac {3 \sqrt {-e} n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^2}{f^{3/2}}-\frac {3 \sqrt {-e} n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^2}{f^{3/2}}+\frac {3 \sqrt {-e} n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^2}{f^{3/2}}-\frac {3 \sqrt {-e} n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^2}{f^{3/2}}-\frac {6 n x \left (a+b \log \left (c x^n\right )\right )^2 b}{f}-\frac {3 \sqrt {-e} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b}{2 f^{3/2}}+\frac {3 \sqrt {-e} n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) b}{2 f^{3/2}}-\frac {3 \sqrt {-e} n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b}{2 f^{3/2}}+\frac {3 \sqrt {-e} n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b}{2 f^{3/2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{f}+\frac {\sqrt {-e} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{2 f^{3/2}}-\frac {\sqrt {-e} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right )}{2 f^{3/2}}\right )\) |
6*a*b^2*n^2*x*Log[d*(e + f*x^2)^m] - 6*b^3*n^3*x*Log[d*(e + f*x^2)^m] + 6* b^3*n^2*x*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 3*b*n*x*(a + b*Log[c*x^n])^2*L og[d*(e + f*x^2)^m] + x*(a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m] - 2*f*m* ((12*a*b^2*n^2*x)/f - (18*b^3*n^3*x)/f + (6*b^2*n^2*(a - b*n)*x)/f - (6*b^ 2*Sqrt[e]*n^2*(a - b*n)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/f^(3/2) + (18*b^3*n^2 *x*Log[c*x^n])/f - (6*b^3*Sqrt[e]*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^ n])/f^(3/2) - (6*b*n*x*(a + b*Log[c*x^n])^2)/f + (x*(a + b*Log[c*x^n])^3)/ f - (3*b*Sqrt[-e]*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(2 *f^(3/2)) + (Sqrt[-e]*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/ (2*f^(3/2)) + (3*b*Sqrt[-e]*n*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqr t[-e]])/(2*f^(3/2)) - (Sqrt[-e]*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/S qrt[-e]])/(2*f^(3/2)) + (3*b^2*Sqrt[-e]*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) - (3*b*Sqrt[-e]*n*(a + b*Log[c*x^n])^2*P olyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(2*f^(3/2)) - (3*b^2*Sqrt[-e]*n^2*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/f^(3/2) + (3*b*Sqrt[-e]*n *(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(2*f^(3/2)) + ((3* I)*b^3*Sqrt[e]*n^3*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/f^(3/2) - ((3*I)* b^3*Sqrt[e]*n^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/f^(3/2) - (3*b^3*Sqrt[- e]*n^3*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) + (3*b^2*Sqrt[-e]*n^2* (a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) + (3*b^...
3.2.12.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[(a + b*Log[c*x^n])^p, x]}, Simp[Log[d*(e + f*x^m)^r] u, x] - Simp[f*m*r Int[x^(m - 1)/(e + f*x^m) u, x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && Inte gerQ[m]
\[\int {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )d x\]
\[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ) \,d x } \]
integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a ^3)*log((f*x^2 + e)^m*d), x)
Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\text {Timed out} \]
Exception generated. \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ) \,d x } \]
Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int \ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]